Super fact 111 : Russel’s Paradox is a logical contradiction discovered in 1901 that showed that the mathematical discipline of “Set Theory” was fundamentally flawed. Mathematicians had naively assumed that any definable property can be used to form a collection (or set) of items, but that is not true. An example of the Paradox is “A male barber shaves all men who do not shave themselves and only men who do not shave themselves. Does he shave himself?” Both “yes” and “no” are impossible answers. That is an example of an impossible set. Set theory needed an exclusion of such impossible sets.
Russell’s paradox is a famous logical contradiction discovered by the philosopher and mathematician Bertrand Russell in 1901. To solve the contradiction, you need to remove the assumption that any property can form a set. In other words, not every set is possible. Basically, self-reference cannot be allowed.
To take the example above “A male barber who shaves all men who do not shave themselves and only men who do not shave themselves.” Is something that cannot exist. If the barber shaves himself then he is shaving someone who shaves himself, which was not allowed. If the barber does not shave himself, then he is not shaving all the men who do not shave themselves. Either way, it does not work. Such a barber cannot exist. In general, you cannot define a set anyway you like.
I consider this a super fact because it shows that contractions can be hidden even in mathematical disciplines, and it is important because you certainly don’t want contradictions hidden in a mathematical or scientific discipline. Contradictions lead to more contradictions and lots of problems.
A Crazy Barber Story Involving Our Children
This happened soon after the September 11 attacks in 2001. In addition to planes crashing into buildings, there were attempts at biological warfare by spreading anthrax through the postal service. This is something we paid special attention to at my work because we were making postal sorting machines. It is also the reason I do not like people who write addresses in cursive.
Anyway, my wife called me at work, and she was very upset because our daughter’s hair was falling out. She touched her hair and it just fell off. She did not know what could be causing her hair to suddenly fall out, but she thought that it might have been biological warfare. I told her to call our doctor who had the good sense of suggesting that perhaps the kids had been playing barbershop. As it turned out they had. Our son confessed to cutting off our daughter’s hair. He had realized that this was bad, so he tried to put her hair back as well as he could. Afterwards, she was walking around with loose hair on top and that’s when my wife found her.
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That’s a cute story about your kids, Thomas. The anthrax in the mail thing was quite worrying at the time. I recently listened to a podcast about it; as I recall, the primary suspect was a scientist who possibly had some mental health problems.
I have to admit complete ignorance about set theory, so this post did remedy that to some extent!
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Thank you so much Audrey. You are right about anthrax scare. I remember that in the end we found out it was likely a scientist with mental health problems. However, at the time we did not know that and at work we had to take all kinds if special precautions and come up with ideas for how to protect people from the possibility of spreading germs and viruses via mail.
I just thought it was funny how we needed to pay attention to the fact that allowing a collection/set to be part of the collection/set itself could lead to a logical disaster and no one had seen it for several decades. Words like “all” and “only” can be dangerous.
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My husband, the engineer, loves bringing up Russel’s Paradox. I know it very well 😊 So funny, about your daughter’s hair, but at the time I can imagine your wife was frantic. Maggie
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Ha ha that is interesting Maggie. You already knew about it. Yes you are right, with all the scary news, and then our daughter’s hair suddenly falling off, she panicked.
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I’ve heard the term “set theory” but knew nothing about it until now. Your barber analogy explained it well! 👌 That’s the second sad hair story I ‘ve read today. What is it about kids and haircuts? 😆
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Thank you so much Debbie. I got the idea for the writing about our kids hairy story and even Russel’s paradox from reading Kymber’s hair story earlier today.
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